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Abstract

An approach to compute the polarizability tensor of magnetic nanoparticles having general
ellipsoidal shape is presented. We find a surprisingly excellent quantitative agreement between
calculated and experimental magneto-optical spectra measured in the polar Kerr configuration
from nickel nanodisks of large size (exceeding 100 nm) with circular and elliptical shape. In
spite of its approximations and simplicity, the formalism presented here captures the essential
physics of the interplay between magneto-optical activity and the plasmonic resonance of the
individual particle. The results highlight the key role of the dynamic depolarization effects
to account for the magneto-optical properties of plasmonic nanostructures.

One should consider also the phase difference due to the incoming light hitting a finite size body. There are several ways to account for this phase difference reported in literature [30, 32, 49]. Although, we verified that inclusion of these corrections have negligible effects, and therefore for sake of clarity we neglect them. We point out, in addition, that for the particular geometry used in our experiments, namely perpendicular incidence over flat disks, this phase difference effects are rigorously zero.

For a proper comparison, it is necessary to establish the association between Dx, Dy and Dz and D|| and D⊥ from Moroz. Based on the definitions of the eccentricities given in the text, our prolate profile is characterized by ax = az < ay, so Dx and Dz are equivalent to D|| and Dy to D⊥, whereas the oblate profile is characterized by ax = az < ay, so that Dx is equivalent this time to D⊥, while Dy and Dz toD||.

This value for the embedding medium refractive index is chosen since the nano-disks embedded in air have one side in contact with the glass substrate. In the calculation we don’t account for the dispersion in the disks size, and we assume that the diameters are the average ones, although the dispersion in diameter can be easily included in Eq. (6) (following Ref. [62]), if required.

Sov. Phys. JTEP (1)

Other (8)

This value for the embedding medium refractive index is chosen since the nano-disks embedded in air have one side in contact with the glass substrate. In the calculation we don’t account for the dispersion in the disks size, and we assume that the diameters are the average ones, although the dispersion in diameter can be easily included in Eq. (6) (following Ref. [62]), if required.

For a proper comparison, it is necessary to establish the association between Dx, Dy and Dz and D|| and D⊥ from Moroz. Based on the definitions of the eccentricities given in the text, our prolate profile is characterized by ax = az < ay, so Dx and Dz are equivalent to D|| and Dy to D⊥, whereas the oblate profile is characterized by ax = az < ay, so that Dx is equivalent this time to D⊥, while Dy and Dz toD||.

One should consider also the phase difference due to the incoming light hitting a finite size body. There are several ways to account for this phase difference reported in literature [30, 32, 49]. Although, we verified that inclusion of these corrections have negligible effects, and therefore for sake of clarity we neglect them. We point out, in addition, that for the particular geometry used in our experiments, namely perpendicular incidence over flat disks, this phase difference effects are rigorously zero.

Figures (8)

Scheme of a general ellipsoid embedded in a non-magnetic host medium. The ellipsoid is
under the influence of an acting field E1, and, due
to the induced dipole moments, the electric field E2
inside it changes.

Numerical calculation of the dynamic and static terms of the depolarization field as a
function of the ellipsoid aspect ratio. The abscissa and ordinate shows the relative
eccentricity in between axes ax and
ay, and between ax, and
az, respectively for (a) Dx, (b)
Dy, (c) Dz,, (d)
Lx, (e) Ly , and (f)
Lz.

(a) Dx, (b) Dy, (c)
Dz,, (d) Lx, (e)
Ly , and (f) Lz. The continuous and
dashed lines correspond to the dynamic and static components for the particular cases of
prolate and oblate spheroids, oriented as pictured in between the plots of the two tensors
elements.

Real (a) and imaginary (b) part of εeffxxfor a system of Ni spheres embedded in air (the filling factor
is 10%), for different values of the particles radius. Real (c) and imaginary (b) parts of
εeffxxfor a system of Ni spheres with radius of 10 nm, embedded in
air, for different values of the filling factor. All the calculations are performed
considering or not the effect of the dynamic term in Eq. (7).

SEM images of the Ni disks with D = 100 nm (a) and with D = 160 nm (b), on glass
substrates, made with Hole Colloidal Mask Lithography technique. The thickness is t = 30 nm.
The filling factor can be estimated to be around 13% in both cases. Experimental (c) and
calculated (d) absorption spectra, defined as 1 – T, where T =
It/I0. In the inset it is shown the extinction efficiency
Qext calculated using the imaginary part of the polarizability tensor elements
related to the two directions considered.

(a) SEM images of the Ni elliptical disks with Dlong = 160, Dshort =
100 nm and t = 30 nm, on glass substrates, made with Hole Colloidal Mask Lithography. It can
be seen that the filling factor is around 2%. (b) Experimental and (c) calculated absorption
spectra, defined as 1 – T, where T = It/I0. In the inset it is
shown the extinction efficiency Qext calculated using the imaginary part of the
polarizability tensor elements related to the two directions considered.

(a) Experimental and (b) calculated Kerr angle in P-MOKE configuration for the Ni
elliptical disks. The calculation is performed for the multilayered system air/effective
medium/glass, where nglass = 1.5. The effective medium film thickness is 30 nm and
the filling factor is f = 2%.